ReliaSoft's Ranking
Method

When analyzing interval data, it is
commonplace to assume that the actual failure time occurred at the
midpoint of the interval. To be more conservative, you can use the
starting point of the interval or you can use the end point of the
interval to be most optimistic.
Weibull++
allows you to employ ReliaSoft's ranking method (RRM) when analyzing
interval data. Using an iterative process, this ranking method is an
improvement over the standard ranking method (SRM). This article
presents an example using the two-parameter Weibull distribution to
illustrate how this method is employed. This method can also be easily
generalized for additional models.

Step-by-Step ExampleConsider the test data shown in Table 1.

Number of Items

Type

Last Inspection

Time

1

Exact Failure

10

1

Right Censored

20

2

Left Censored

0

30

2

Exact Failure

40

1

Exact Failure

50

1

Right Censored

60

1

Left Censored

0

70

2

Interval Failure

20

80

1

Interval Failure

10

85

1

Left Censored

0

100

Table 1: Test data.

As a preliminary step, we need to provide
a crude estimate of the Weibull parameters for this data. To begin, we
will extract the exact times-to-failure (10, 40 and 50) and append them
to the midpoints of the interval failures: 50 (for the interval of 20 to
80) and 47.5 (for the interval of 10 to 85). Now, the extracted list
consists of the data in Table 2.

Number of Items

Type

Last Inspection

Time

1

Exact
Failure

10

2

Exact
Failure

40

1

Exact
Failure

47.5

3

Exact
Failure

50

Table 2: Union of exact
times-to-failure with the midpoint of the interval failures.

Using the traditional rank regression,
we obtain the first initial estimates:

Step 1For all intervals, we obtain a
weighted "midpoint" using:

This transforms the data into the format
displayed in Table 3.

Number of Items

Type

Last Inspection

Time

Weighted Midpoint

1

Exact
Failure

10

2

Exact
Failure

40

1

Exact
Failure

50

2

Interval Failure

20

80

42.837

1

Interval Failure

10

85

39.169

Table 3: Union of exact
times-to-failure with the midpoint based on parameters β
and η.

Step 2Now arrange the data, as shown in
Table 4.

Number of Items

Time

1

10

1

39.169

2

40

2

42.837

1

50

Table 4: Union of exact
times-to-failure in ascending order.

Step 3Now consider the left and right
censored data shown in Table 5.

Table 5: Computation of
increments for computing a revised mean order number.

In general, for left censored data:

The increment term for n left censored
items at time = t0, with a time-to-failure of ti,
when t0
ti-1 is zero.

When t0 > ti-1
the contribution is:

Or:

Where ti-1 is the
time-to-failure previous to the ti
time-to-failure and n is the number of units associated with that
time-to-failure (or units in the group).

Or:

Step 4Sum up the increments
(horizontally in rows) as shown in Table 6.

Table 6: Increments
solved numerically.

Step 5Compute new mean order numbers (MON), as shown in Table 7, utilizing
the increments obtained in Table 6, by adding the "number of items" plus
the "previous MON" plus the current "increment."

Table 7 - Mean Order
Numbers (MON).

Step 6Compute the median ranks based on these new MONs, as shown in Table 8.

Table 8 - Mean Order
Numbers with ranks for a sample size of 13 units.

Step 7Compute new Beta and Eta using standard rank regression and based upon
the data as shown in Table 9.

Table 9 - New times and
median ranks for regression.

Step 8Return and repeat the process
from Step 1 until an acceptable convergence is reached on the parameters
(i.e. until the parameter values stabilize).

ResultsThe results of the first five
iterations are shown in Table 10.